Variational R-matrix method for quantum tunneling problems

American Journal of Physics

Volumn 71, Issue 3, 2003, Pages 273-278

  Le Rouzo, Hervé ^a  

^a CNRS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037339540     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1519231     Document Type: Article

References (31)

1. H. Ness and A. J. Fisher, "Dynamical effective potential for tunneling: an exact method and a path-integral technique," Appl. Phys. A: Mater. Sci. Process. 66, S919-S923 (1998); Y. Ko, M. Shin, J. S. Ha, and K. W. Park, "Green-function calculations of coherent electron transport in a gated Si nanowire," ET J. 22, 19-26 (2000); M. Tsukada, N. Kobayashi, M. Brandbyge, and S. Nakanishi, "Physics of artificial nano-structures on surfaces," Prog. Surf. Sci. 64, 139-155 (2000); I. P. Batra, P. Sen, and S. Ciraci, "Quantum transport through one-dimensional aluminium wires," J. Vac. Sci. Technol. B 20, 812-817 (2002).
2. H. Ness and A. J. Fisher, "Dynamical effective potential for tunneling: an exact method and a path-integral technique," Appl. Phys. A: Mater. Sci. Process. 66, S919-S923 (1998); Y. Ko, M. Shin, J. S. Ha, and K. W. Park, "Green-function calculations of coherent electron transport in a gated Si nanowire," ET J. 22, 19-26 (2000); M. Tsukada, N. Kobayashi, M. Brandbyge, and S. Nakanishi, "Physics of artificial nano-structures on surfaces," Prog. Surf. Sci. 64, 139-155 (2000); I. P. Batra, P. Sen, and S. Ciraci, "Quantum transport through one-dimensional aluminium wires," J. Vac. Sci. Technol. B 20, 812-817 (2002).
3. H. Ness and A. J. Fisher, "Dynamical effective potential for tunneling: an exact method and a path-integral technique," Appl. Phys. A: Mater. Sci. Process. 66, S919-S923 (1998); Y. Ko, M. Shin, J. S. Ha, and K. W. Park, "Green-function calculations of coherent electron transport in a gated Si nanowire," ET J. 22, 19-26 (2000); M. Tsukada, N. Kobayashi, M. Brandbyge, and S. Nakanishi, "Physics of artificial nano-structures on surfaces," Prog. Surf. Sci. 64, 139-155 (2000); I. P. Batra, P. Sen, and S. Ciraci, "Quantum transport through one-dimensional aluminium wires," J. Vac. Sci. Technol. B 20, 812-817 (2002).
4. H. Ness and A. J. Fisher, "Dynamical effective potential for tunneling: an exact method and a path-integral technique," Appl. Phys. A: Mater. Sci. Process. 66, S919-S923 (1998); Y. Ko, M. Shin, J. S. Ha, and K. W. Park, "Green-function calculations of coherent electron transport in a gated Si nanowire," ET J. 22, 19-26 (2000); M. Tsukada, N. Kobayashi, M. Brandbyge, and S. Nakanishi, "Physics of artificial nano-structures on surfaces," Prog. Surf. Sci. 64, 139-155 (2000); I. P. Batra, P. Sen, and S. Ciraci, "Quantum transport through one-dimensional aluminium wires," J. Vac. Sci. Technol. B 20, 812-817 (2002).
5. V. E. Barlette, M. M. Leite, and S. K. Adhikari, "Integral equations of scattering in one dimension," Am. J. Phys. 69, 1110-1113 (2001).
6. E. P. Wigner and L. Eisenbud, "Higher angular momenta and long range interaction in resonance reactions," Phys. Rev. 72, 29-41 (1947). See also the celebrated review by A. M. Lane and R. G. Thomas, "R-matrix theory of nuclear reactions," Rev. Mod. Phys. 30, 257-353 (1958).
7. E. P. Wigner and L. Eisenbud, "Higher angular momenta and long range interaction in resonance reactions," Phys. Rev. 72, 29-41 (1947). See also the celebrated review by A. M. Lane and R. G. Thomas, "R-matrix theory of nuclear reactions," Rev. Mod. Phys. 30, 257-353 (1958).
8. 2: An R-matrix calculation," Phys. Rev. A 13, 2049-2056 (1976).
9. 2: An R-matrix calculation," Phys. Rev. A 13, 2049-2056 (1976).
10. L. Smrcka, "R-matrix and the coherent transport in mesoscopic systems," Superlattices Microstruct. 8, 221-224 (1990).
11. W. Kohn, "Variational methods in nuclear collision problems," Phys. Rev. 74, 1763-1772 (1948).
12. R. K. Nesbet, Variational Methods in Electron-Atom Scattering Theory (Plenum, New York, 1980).
13. G. Raseev and H. Le Rouzo, "A continuum spectrum multichannel finite-volume variational method," Proceedings of the XIII ICPEAC, Berlin, 1983, p. 36; H. Le Rouzo and G. Raseev, "Finite-volume variational method: first application to direct molecular photoionisation," Phys. Rev. A 29, 1214-1223 (1984).
14. G. Raseev and H. Le Rouzo, "A continuum spectrum multichannel finite-volume variational method," Proceedings of the XIII ICPEAC, Berlin, 1983, p. 36; H. Le Rouzo and G. Raseev, "Finite-volume variational method: first application to direct molecular photoionisation," Phys. Rev. A 29, 1214-1223 (1984).
15. G. H. Greene, "Atomic photoionization in a strong magnetic field," Phys. Rev. A 28, 2209-2216 (1983).
16. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
17. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
18. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
19. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
20. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
21. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
22. 2," J. Phys. B 18, 423-439 (1985); C. H. Greene and L. Kim, "Two-electron excitations in atomic calcium," Phys. Rev. A 36, 2706-2717 (1987); C. H. Greene and M. Aymar, "Spin-orbit effects in the heavy alkaline-earth atoms," ibid. 44, 1773-1790 (1991); I. Wilhelmy, L. Ackermann, A. Görling, and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. I. Valence photoionization," J. Chem. Phys. 100, 2808-2820 (1994); I. Wilhelmy and N. Rösch, "Molecular photoionization cross sections by the Lobatto technique. II. Core level photoionization," Chem. Phys. 185, 317-332 (1994).
23. P. Hamacher and J. Hinze, "Finite-volume variational method for the Dirac equation," Phys. Rev. A 44, 1705-1711 (1991); R. Szmytkowski, "Variational R-matrix methods for many-electron systems: Unified relativistic theory," ibid. 63, 062704-1-062704-14 (2001).
24. P. Hamacher and J. Hinze, "Finite-volume variational method for the Dirac equation," Phys. Rev. A 44, 1705-1711 (1991); R. Szmytkowski, "Variational R-matrix methods for many-electron systems: Unified relativistic theory," ibid. 63, 062704-1-062704-14 (2001).
25. T. Pang, "A numerical method for quantum tunneling," Comput. Phys. 9, 602-605 (1995). See also T. Pang, An Introduction to Computational Physics (Cambridge University Press, New York, 1997).
26. T. Pang, "A numerical method for quantum tunneling," Comput. Phys. 9, 602-605 (1995). See also T. Pang, An Introduction to Computational Physics (Cambridge University Press, New York, 1997).
27. L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Mechanics with Applications to Chemistry (Dover, New York, 1963), p. 169, Eq. (24-15).
28. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1961), Vol. I.
29. 1x) has unit amplitude.
30. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II, pp. 1650-1660.
31. In atomic units: m = h = |e| = 1. Atomic units of length and energy are 1 bohr = 0.529 177 Å and 1 hartree = 27.211 65 eV, respectively.